The angle between the pair of striaght lines `y^(2)sin^(2)theta-xysin^(2)theta+x^(2)(cos^(2)theta-1)=0` is

Options

A. `pi//3`
B. `pi//4`
C. `2pi//3`
D. none of these

Answer

To find the angle between the pair of straight lines given by the equation

\[ y^2 \sin^2 \theta - xy \sin^2 \theta + x^2 (\cos^2 \theta - 1) = 0, \]

we can follow these steps:

### Step 1: Identify the coefficients
The given equation can be compared with the general form of a pair of straight lines:

\[ ax^2 + 2hxy + by^2 = 0. \]

Here, we identify:
- \( a = \cos^2 \theta - 1 \)
- \( b = \sin^2 \theta \)
- \( 2h = -\sin^2 \theta \)

### Step 2: Calculate the angle between the lines
The angle \( \phi \) between the two lines can be calculated using the formula:

\[ \tan \phi = \frac{2h}{a + b}. \]

### Step 3: Substitute the values of \( a \), \( b \), and \( h \)
Substituting the identified coefficients into the formula:

\[ \tan \phi = \frac{-\sin^2 \theta}{(\cos^2 \theta - 1) + \sin^2 \theta}. \]

### Step 4: Simplify the denominator
Now simplify the denominator:

\[ \cos^2 \theta - 1 + \sin^2 \theta = \cos^2 \theta + \sin^2 \theta - 1 = 1 - 1 = 0. \]

### Step 5: Analyze the result
Since the denominator is zero, this implies that \( \tan \phi \) is undefined, which indicates that the angle \( \phi \) is \( 90^\circ \). Therefore, the lines are perpendicular.

### Final Answer
The angle between the pair of straight lines is \( 90^\circ \).

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